Goal revealed last week that the Serie A side were plotting an initial bid for the wantaway Foxes star of around €35 million (£30.7m), with new sporting director Monchi having already made contact with the player's agent.
Tottenham are also reported to be interested in Mahrez, as are north London rivals Arsenal, though Leicester will have to lower their asking price if either Premier League club are to complete a deal for the winger.
Goal understands the Algeria international and his team have agreed terms with Roma, however, and are simply waiting for negotiations between the clubs to play out.
Foxes boss Shakespeare previously denied there had been an official offer for Mahrez, who has made it publicly clear he is looking for a move away from the King Power, though has now revealed the club have rejected an offer from the Giallorossi.
"I think I was quoted as saying on the last press day that there had been no bids," Shakespeare told the media ahead of the Foxes' meeting with Liverpool in Hong Kong. "There was a bid from Roma I was told after, I'll put the record straight on that one.
"It was politely declined on the basis that it was a low offer.
"What that offer was, don't ask me because I don't know and I don't get involved in that.
"I think you saw in Riyad's performance, and I've seen a lot of quotes from our own players since, he is a Leicester player and until we're told otherwise we expect him to perform to the high standards we all set.
"He's a team player. Individuals are involved in this team sport so they have to adhere to the rules of the team game. While he does that he'll be considered for selection."
Asked whether Roma's bid was the only one the club had received thus far, Shakespeare replied simply: "Yes."